Prescribed Mean Curvature Min-Max Theory in Some Non-Compact Manifolds

TL;DR

This paper introduces a new min-max technique for finding hypersurfaces with prescribed mean curvature in certain non-compact manifolds, with applications to asymptotically flat spaces and Euclidean settings.

Contribution

It develops a novel prescribed mean curvature min-max method applicable to non-compact manifolds, enabling existence results for hypersurfaces with specific curvature properties.

Findings

Existence of hypersurfaces with prescribed mean curvature in Euclidean space under certain conditions.

Existence of constant mean curvature surfaces in asymptotically flat 3-manifolds.

Extension of min-max theory to non-compact settings.

Abstract

This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension $3\le n+1 \le 7$ and consider a smooth function $h\colon \mathbb{R}^{n+1}\to \mathbb{R}$ which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on $h$, there exists a closed hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$ with mean curvature prescribed by $h$. Second, let $(M^3,g)$ be an asymptotically flat 3-manifold and fix a constant $c > 0$. We show that, under an additional assumption on $M$, it is possible to find a closed surface $\Sigma$ of constant mean curvature $c$ in $M$.